Designing A Folded Sheet Object

ABSTRACT

It is provided a computer-implemented method for designing a folded sheet object, comprising the steps of providing (S 10 ) panels separated by bend lines, including at least four adjacent panels forming a cycle and separated by concurrent bend lines, with predetermined angles between successive bend lines; and determining (S 20 ) a control law linking the angles between the adjacent panels of the cycle, as a function of the predetermined angles between successive bend lines. Such a method improves the design of a folded sheet object.

RELATED APPLICATION(S)

This application claims priority under 35 U.S.C. §119 or 365 to Europe,Application No. 13305841.2, filed Jun. 20, 2013. The entire teachings ofthe above application(s) are incorporated herein by reference.

FIELD OF THE INVENTION

The invention relates to the field of computer programs and systems, andmore specifically to a method, system and program for designing a foldedsheet object, as well as a folded sheet object obtainable by said methodand a data file storing said folded sheet object.

BACKGROUND OF THE INVENTION

A number of systems and programs are offered on the market for thedesign, the engineering and the manufacturing of objects. CAD is anacronym for Computer-Aided Design, e.g. it relates to software solutionsfor designing an object. CAE is an acronym for Computer-AidedEngineering, e.g. it relates to software solutions for simulating thephysical behavior of a future product. CAM is an acronym forComputer-Aided Manufacturing, e.g. it relates to software solutions fordefining manufacturing processes and operations. In such systems, thegraphical user interface (GUI) plays an important role as regards theefficiency of the technique. These techniques may be embedded withinProduct Lifecycle Management (PLM) systems. PLM refers to a businessstrategy that helps companies to share product data, apply commonprocesses, and leverage corporate knowledge for the development ofproducts from conception to the end of their life, across the concept ofextended enterprise.

The PLM solutions provided by Dassault Systèmes (under the trademarksCATIA, ENOVIA and DELMIA) provide an Engineering Hub, which organizesproduct engineering knowledge, a Manufacturing Hub, which managesmanufacturing engineering knowledge, and an Enterprise Hub which enablesenterprise integrations and connections into both the Engineering andManufacturing Hubs. All together the system delivers an open objectmodel linking products, processes, resources to enable dynamic,knowledge-based product creation and decision support that drivesoptimized product definition, manufacturing preparation, production andservice.

Some CAD systems now deal with packaging design. Packaging designsystems may typically allow to define the shape of a flat carton boardsheet and the arrangement of bend lines on said sheet in such a waythat, according to a folding procedure, which may also be a matter ofpackaging design, the flat sheet changes itself into a three dimensionalbox that is able to contain some kind of goods. Packaging design CADsystems include for example ArtiosCAD software (registered trademark).ArtiosCAD includes geometric tools for a user to interactively definethe outside shape, bend lines and panels of a flat carton board sheet aswell as the folding procedure and the resulting three-dimensional shapeof the package. For the sake of comprehensiveness, it is noted thatArtiosCAD also includes other functions related to packaging design(drawing, artwork, flexible package, layout for manufacturing) that areout of the scope of the present discussion.

FIG. 1 illustrates the typical flat view of a three dimensional box,carton board sheet 100, designed by ArtiosCAD. Dotted lines representbend lines. Solid lines represent boundaries of the planar carton boardsheet. Portions of sheet 100 bounded by dotted and/or solid lines arenamed panels. By definition, two panels are said to be adjacent if theyshare a bend line. Panels are numbered from 1 to 13 in the figures. FIG.2 illustrates three-dimensional box 200 obtained by folding carton boardsheet 100 of FIG. 1. Only visible panels of FIG. 1 are tagged withnumbers on FIG. 2.

The key point of state of the art packaging design systems is related tothe bend lines arrangement: each bend line can be folded independently.Formally, this means that the panels' adjacency graph is acyclic. Thepanels' adjacency graph is a non-directed graph defined as follows.Vertices are panels' numbers and arcs capture the adjacencyrelationship: an arc joins panel number i and panel number j if the saidpanels are adjacent (meaning that they share a bend line). From themathematical point of view, the flat carton board sheet corresponds to aplanar graph and the panels' adjacency graph is its dual graph. FIG. 3illustrates panels' adjacency graph 300 of carton board sheet 100 ofFIG. 1. It is clearly acyclic. The technical consequence is that thefolding procedure can be directly computed by choosing a base panel onthe flat sheet and by traversing the acyclic graph.

As opposed to the acyclic graph, the theoretical case of a local cyclein the panels' adjacency is not handled by the prior art in asatisfactory way. The corresponding significant folding situation ischaracterized when at least four bend lines are concurrent at a point Pand when no boundary line meets point P. It is well known from state ofthe art that folding is generally impossible with three or lessconcurrent bend lines. Furthermore, it is well known from state of theart that folding is generally impossible when bend lines in such a cycleare not concurrent and not parallel (parallelism is discussed later).

FIG. 4 illustrates sheet 400 having seven panels including four panelssharing concurrent bend lines (at point P) in a cyclic way. Thecorresponding adjacency graph 500 shown on FIG. 5 includes the cycle ofarcs (1,2), (2,3), (3,4) and (4,1) caused by four concurrent bend lines.Clearly, panels adjacent to concurrent bend lines cannot be foldedindependently. For example, choose panel 1 as the fixed panel and rotatepanel 4 upward around the bend line shared with panel 1. This causespanels 2 and 3 to move accordingly. The final position illustrated inFIG. 6 is such that panels 1 and 2 are in the same plane. Panels 5, 6and 7 are folded independently. It must be understood that the cyclicsituation and the folding conditions are not altered if carton boardsheet 400 is punched by a hole in the neighborhood of point P, meaningthat there is no material around point P. In particular, the panels'adjacency graph is unchanged. FIGS. 7-8 illustrate this situation withhole 700 around point P on sheet 750, which is otherwise the same assheet 400 of FIG. 4.

As mentioned previously, a cyclic situation with folding is allowed byparallel bend lines, as illustrated by FIGS. 9-11 which respectivelyshow the sheet when unfolded, the panel's adjacency graph, and the sheetafter folding. The folding is made possible because the distanceseparating bend lines of panel 2 is equal to the distance separatingbend lines of panel 4. This situation is out of the scope of the presentdiscussion because it is not difficult to compute.

In this context, the invention aims at improving the design of a foldedsheet object.

SUMMARY OF THE INVENTION

According to one aspect, it is therefore provided a computer-implementedmethod for designing a folded sheet object. The method comprises thestep of providing panels separated by bend lines, including at leastfour adjacent panels forming a cycle and separated by concurrent bendlines, with predetermined angles between successive bend lines. And themethod also comprises the step of determining a control law linking theangles between the adjacent panels of the cycle, as a function of thepredetermined angles between successive bend lines

The method may comprise one or more of the following:

-   -   the control law fixes, given predetermined values for a number        of the angles between pairs of the adjacent panels equal to the        degree of freedom of the cycle, values for the angles between        other pairs of the adjacent panels;    -   when the cycle has four adjacent panels, the control law        comprises the solution of an equation of the type F(ψ, φ)=0,        where ψ and φ are the angles between successive pairs of the        adjacent panels and φ is predetermined, the angles between the        two other pairs of adjacent panels of the cycle stemming from        said solution;    -   the equation is

F(ψ,φ)=k ₁ −k ₂ cos ψ+k ₃ cos ψ cos φ+k ₄ cos φ−sin ψ sin φ

with:

$k_{1} = \frac{{\cos \; \alpha_{1}\cos \; \alpha_{2}\cos \; \alpha_{4}} - {\cos \; \alpha_{3}}}{\sin \; \alpha_{2}\sin \; \alpha_{4}}$k₂ = cot  α₄sin  α₁ k₃ = cos  α₁ k₄ = cot  α₂sin  α₁

and α_(i(i=1 . . . 4)) are the predetermined angles (α_(i)) betweensuccessive bend lines, where ψ is the angle between the pair of theadjacent panels corresponding to α₁ and α₂, and φ is the angle betweenthe pair of the adjacent panels corresponding to α₁ and α₃;

-   -   when the cycle has more than four panels, determining the        control law comprises partitioning the panels into four groups,        and for each group which comprises at least two panels,        determining a virtual panel that forms a cycle with the panels        of the group, thereby forming a virtual cycle of four adjacent        panels that each correspond to a respective group, and the        control law comprises the solution of equation F defined for the        virtual cycle; and/or    -   when the cycle has more than four panels and a degree of freedom        higher or equal to 2, at least one angle between a pair of        adjacent panels is controlled by a variation law depending on        the angle between another pair of adjacent panels.

It is further proposed a folded sheet object obtainable by the abovemethod.

It is further proposed a data file storing said folded object.

It is further proposed a computer-implemented method for simulating thefolding of a folded sheet object. The method comprises the step ofproviding predetermined values for a number of the angles between pairsof the adjacent panels. The method also comprises the step ofdetermining values of the other angles between pairs of the adjacentpanels with the control law.

It is further proposed a computer program comprising instructions forperforming any or both of the above methods. The computer program isadapted to be recorded on a computer readable storage medium.

It is further proposed a computer readable storage medium havingrecorded thereon the above computer program.

It is further proposed a CAD system comprising a processor coupled to amemory and a graphical user interface, the memory having recordedthereon the above computer program.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing will be apparent from the following more particulardescription of example embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingembodiments of the present invention.

Embodiments of the invention will now be described, by way ofnon-limiting example, and in reference to the accompanying drawings,where:

FIGS. 1-11 illustrate the context of the methods;

FIG. 12 shows a flowchart of an example of the design method;

FIG. 13 shows a flowchart of an example of the simulation method;

FIG. 14 shows an example of a graphical user interface;

FIG. 15 shows an example of a client computer system; and

FIGS. 16-48 illustrate examples of the methods.

DETAILED DESCRIPTION OF THE INVENTION

A description of example embodiments of the invention follows.

FIG. 12 shows a flowchart of an example of computer-implemented methodfor designing a folded sheet object. The design method comprisesproviding S10 panels separated by bend lines, including at least fouradjacent panels forming a cycle and separated by concurrent bend lines,with predetermined angles between successive bend lines. And the designmethod also comprises determining S20 a control law linking the anglesbetween the adjacent panels of the cycle, as a function of thepredetermined angles between successive bend lines. Such a methodimproves the design of a folded sheet object.

The design method considers the case mentioned earlier of at least fouradjacent panels forming a cycle separated by concurrent bend lines. Thedesign method then determines at S20 a control law that allows thepotential simulation of the folding of a folded sheet, since the controllaw links the angles between the adjacent panels of the cycle. Thecontrol law is a function of the predetermined angles between successivebend lines, such that the control law may be determined once and for allafter the providing S10. Thus the user need not intervene for thesimulation. For this reason, the design method improves the design of afolded sheet object, by considering a specific case (i.e. at least fouradjacent panels forming a cycle and separated by concurrent bend lines)that may occur in some industrial cases (e.g. the packaging industry)and adding to the folded sheet object some data (i.e. the control law)that may conveniently be used for simulation of the folding of thefolded sheet object, without the need for user intervention.

State of the art packaging design CAD systems cannot handle cyclicsituations featuring concurrent bend lines. The background artlimitation is the acyclic adjacency graph, which is unable to handlecomplex cyclic folding. However, complex cyclic folding is useful inpackaging design because it helps several panels to be folded at thesame time: folding one panel involves neighboring panels folding becauseof the cyclic dependency. Conversely, an acyclic procedure requires eachpanel to be folded individually. Folding several panels at the same timethrough cyclic dependency simplifies the physical folding machine, whichin turn shortens the time to design, build and repair the foldingmachine and, finally, shortens the time to manufacture packaged goods.Some prior art packaging design systems tolerate a cyclic adjacencygraph when dealing with the flat sheet design. But, when creating thefolding procedure (in simulation), the user is asked to break the cycleby selecting a bend line. This action is not legitimate design intent.It is motivated by the weakness of such prior packaging design system.The resulting folding simulation is unrealistic and time consuming sincethe user is asked again to manage by hand the behavior of folded panelsin the neighborhood of the cut bend line.

On the contrary, the simulation of the folding of a folded sheet objectdesigned by the present design method may be automatic, excluding anyuser action, or any action that consists in breaking the cycle ofadjacent panels (other actions may be performed by the user, e.g. todefine the folding configuration, or to define physicalconstraints/variation laws when there are more than four panels, asexplained later).

A folded sheet object is a specific kind of modeled object. A modeledobject is any object defined/described by structured data that may bestored in a data file (i.e. a piece of computer data having a specificformat) and/or on a memory of a computer system. A folded sheet objectis a modeled object that represents any type of physical sheet that maybe folded. For example, the folded sheet object designed may represent aplanar sheet (the folded sheet object may thus have a “planar state”)that may form any kind of package after folding. By extension, theexpression “folded sheet object” may designate the data itself, or thephysical sheet. Notably, a folded sheet object may comprise dataallowing the simulation of the folding of the folded sheet object(thereby representing the folding of the physical sheet represented bythe folded sheet object). The folded sheet object obtained by the methodhas a specific structure. The data forming the folded sheet objectdesigned by the method comprise any data that describe at least fouradjacent panels forming a cycle, bend lines separating the panels,predetermined angles between successive bend lines, and eventually thecontrol law.

The design method is for designing a folded sheet object, e.g. the stepsof the design method constituting at least some steps of such design.“Designing a folded sheet object” designates any action or series ofactions which is at least part of a process of elaborating a foldedsheet object. Thus, the design method may comprise creating the foldedsheet from scratch, before the providing S10. Alternatively, the designmethod may comprise providing a folded sheet object previously created,and then modifying the folded sheet object, e.g. for adding to it thecontrol law determined at S20.

The simulation of the folding of such a folded sheet object is easy, asillustrated with reference to FIG. 13. Simulating the folding of afolded sheet object is any action that is, at least part of, describingand/or representing a physical folding of the physical sheet representedby the folded sheet object. The simulation of FIG. 13 comprises a stepof providing S30 predetermined values for a number N of the anglesbetween pairs of the adjacent panels. In other words, the simulation ofthe folding starts with specific objective values for some of the anglesbetween adjacent panels (i.e. angles between two planes intersecting onthe bend line separating the adjacent panels, e.g. taken on the side ofthe planes occupied by the panels), such that the simulation may thencomprise a step of determining S40 values of the other angles betweenpairs of the adjacent panels, such values being compliant with thecontrol law. The determining S40 is done easily with the control law,since the control laws allows, by definition, the deduction of at leastone value, for each of the remaining angles, from the angles for whichvalues were given, with a degree of freedom in the determining S40 thatdepends on N and the total number of panels, and/or potentialconstraints provided for the angles between adjacent panels. This isexplained later.

The determining S40 may be automated. The values provided at theproviding step S30 may be provided by the user, or by an underlyingprocess that defines constraints on some of the panels (e.g. to simulatea physical pressure in order to perform the folding of the physicalsheet represented by the folded sheet object), in which latter case theproviding S30 and the determining S40 may be iterated to simulate thewhole folding process of the folded sheet object, e.g. withpredetermined time steps. Such a simulation may be subsequent to thedesign of the modeled object, and/or it may be performed at a separatetime, based on the folded sheet object designed by the design method.

The folded sheet object may be a CAD modeled object or a part of a CADmodeled object. In any case, the folded sheet object designed by themethod may represent the CAD modeled object or at least part of it, e.g.a 3D space occupied by the CAD modeled object. A CAD modeled object isany object defined by data stored in a memory of a CAD system. Accordingto the type of the system, the modeled objects may be defined bydifferent kinds of data. A CAD system is any system suitable at leastfor designing a modeled object on the basis of a graphicalrepresentation of the modeled object, such as CATIA. Thus, the datadefining a CAD modeled object comprise data allowing the representationof the modeled object (e.g. geometric data, for example includingrelative positions in space).

The design and/or simulation method may be included in a manufacturingprocess, which may comprise, after performing the method, producing aphysical product corresponding to the folded sheet object, e.g.performing a packaging method with a physical sheet represented by thefolded sheet object, e.g. the folding in the packaging method beingperformed according to the simulation method. In any case, the foldedsheet object designed by the method may represent a manufacturingobject, e.g. a physical sheet such as a board sheet in any foldablematerial, e.g. carton/paper/plastic/metal. The method can be implementedusing a CAM system, such as DELMIA. A CAM system is any system suitableat least for defining, simulating and controlling manufacturingprocesses and operations.

The methods are computer-implemented. This means that the methods areexecuted on at least one computer, or any system alike. For example, themethods may be implemented on a CAD system. Thus, steps of the methodsare performed by the computer, possibly fully automatically, or,semi-automatically (e.g. steps which are triggered by the user and/orsteps which involve user-interaction, for example for validatingresults).

A typical example of computer-implementation of the design and/orsimulation method is to perform the method with a system suitable forthis purpose. The system may comprise a memory having recorded thereoninstructions for performing the method. In other words, software isalready ready on the memory for immediate use. The system is thussuitable for performing the method without installing any othersoftware. Such a system may also comprise at least one processor coupledwith the memory for executing the instructions. In other words, thesystem comprises instructions coded on a memory coupled to theprocessor, the instructions providing means for performing the method.Such a system is an efficient tool for designing a folded sheet object.

Such a system may be a CAD system. The system may also be a CAE and/orCAM system, and the CAD modeled object may also be a CAE modeled objectand/or a CAM modeled object. Indeed, CAD, CAE and CAM systems are notexclusive one of the other, as a modeled object may be defined by datacorresponding to any combination of these systems.

The system may comprise at least one GUI for launching execution of theinstructions, for example by the user. Notably, the GUI may allow theuser to trigger the step of providing S10 or S30, and then, if the userdecides to do so, e.g. by launching a specific function, to trigger therest of the method.

The folded sheet object may be 3D (i.e. three-dimensional). This meansthat the folded sheet object is defined by data allowing its 3Drepresentation. A 3D representation allows the viewing of therepresentation from all angles. For example, the folded sheet object,when 3D represented, for example during simulation of the folding, maybe handled and turned around any of its axes, or around any axis in thescreen on which the representation is displayed. This notably excludes2D icons, which are not 3D modeled, even when they represent somethingin a 2D perspective. The display of a 3D representation facilitatesdesign or understanding of a simulation (i.e. increases the speed atwhich designers or engineers statistically accomplish their task). Thisspeeds up the manufacturing process in the industry.

FIG. 14 shows an example of the GUI of a typical CAD system, which mayinclude functionality for performing the design or the simulation.Standard functionalities of such a system are now discussed.

The GUI may be a typical CAD-like interface, having standard menu bars2110, 2120, as well as bottom and side toolbars 2140, 2150. Such menuand toolbars contain a set of user-selectable icons, each icon beingassociated with one or more operations or functions, as known in theart. Some of these icons are associated with software tools, adapted forediting and/or working on the 3D modeled object 2000 (folded sheetobject) displayed in the GUI. The software tools may be grouped intoworkbenches. Each workbench comprises a subset of software tools. Inparticular, one of the workbenches is an edition workbench, suitable forediting geometrical features of the modeled product 2000. In operation,a designer may for example pre-select a part of the object 2000 and theninitiate an operation (e.g. a sculpting operation, or any otheroperation such as a change of dimension, color, etc.) or editgeometrical constraints by selecting an appropriate icon. For example,typical CAD operations are the modeling of the punching or the foldingof the 3D modeled object displayed on the screen. The GUI may forexample display data 2500 related to the displayed product 2000. In theexample of FIG. 14, the data 2500, displayed as a “feature tree”, andtheir 3D representation 2000 pertain to a brake assembly including brakecaliper and disc. The GUI may further show various types of graphictools 2130, 2070, 2080 for example for facilitating 3D orientation ofthe object, for triggering a simulation of an operation of an editedproduct or rendering various attributes of the displayed product 2000. Acursor 2060 may be controlled by a haptic device to allow the user tointeract with the graphic tools.

FIG. 15 shows an example of the architecture of the system as a clientcomputer system, e.g. a workstation of a user.

The client computer comprises a central processing unit (CPU) 1010connected to an internal communication BUS 1000, a random access memory(RAM) 1070 also connected to the BUS. The client computer is furtherprovided with a graphics processing unit (GPU) 1110 which is associatedwith a video random access memory 1100 connected to the BUS. Video RAM1100 is also known in the art as frame buffer. A mass storage devicecontroller 1020 manages accesses to a mass memory device, such as harddrive 1030. Mass memory devices suitable for tangibly embodying computerprogram instructions and data include all forms of nonvolatile memory,including by way of example semiconductor memory devices, such as EPROM,EEPROM, and flash memory devices; magnetic disks such as internal harddisks and removable disks, magneto-optical disks, and CD-ROM disks 1040.Any of the foregoing may be supplemented by, or incorporated in,specially designed ASICs (application-specific integrated circuits). Anetwork adapter 1050 manages accesses to a network 1060. The clientcomputer may also include a haptic device 1090 such as a cursor controldevice, a keyboard or the like. A cursor control device is used in theclient computer to permit the user to selectively position a cursor atany desired location on screen 1080. By screen, it is meant any supporton which displaying may be performed, such as a computer monitor. Inaddition, the cursor control device allows the user to select variouscommands, and input control signals. The cursor control device includesa number of signal generation devices for input control signals tosystem. Typically, a cursor control device may be a mouse, the button ofthe mouse being used to generate the signals.

To cause the system to perform the method, it is provided a computerprogram comprising instructions for execution by a computer, theinstructions comprising means for this purpose. The program may forexample be implemented in digital electronic circuitry, or in computerhardware, firmware, software, or in combinations of them. Apparatus ofthe invention may be implemented in a computer program product tangiblyembodied in a machine-readable storage device for execution by aprogrammable processor; and method steps of the invention may beperformed by a programmable processor executing a program ofinstructions to perform functions of the invention by operating on inputdata and generating output. The instructions may advantageously beimplemented in one or more computer programs that are executable on aprogrammable system including at least one programmable processorcoupled to receive data and instructions from, and to transmit data andinstructions to, a data storage system, at least one input device, andat least one output device. The application program may be implementedin a high-level procedural or object-oriented programming language, orin assembly or machine language if desired; and in any case, thelanguage may be a compiled or interpreted language. The program may be afull installation program, or an update program. In the latter case, theprogram updates an existing CAD system to a state wherein the system issuitable for performing the method.

The design method is now further discussed.

As explained earlier, the folded sheet object is provided at S10 as datadescribing panels (representing physical panels, e.g. planar) separatedby bend lines (representing bend lines between two physical panels, e.g.straight). Each bend line may be associated to two panels separated bysaid bend line. Such data may be provided at S10 as geometry andtopology. The data may comprise or not comprise information regardingthe thickness of the panels of the folded sheet object. In an example,the method is performed as if the physical sheet is made of a zerothickness material. The panels may have any form. The panels may bequadrilateral, e.g. rectangular as on FIG. 4 or trapezoid as on FIG. 1,as such panels are the most practical ones in the packaging industry.The folded sheet object may or may not comprise holes, as on FIG. 7.

Now, the folded sheet object provided at S10 is of a specific kind, asit includes at least four adjacent panels forming a cycle and separatedby concurrent bend lines, as sheet 400 of FIG. 4 or sheet 750 of FIG. 7.In other words, the folded sheet object provided at S10 includes a setof panels (with a number equal or higher to 4) that are two-by-twoadjacent and form a cycle. A cycle designates the fact that browsing thepanels of the cycle from one panel to an adjacent one, starting from aninitial panel of the cycle, eventually brings back to the initial panel.For example, an adjacency graph may be provided at S10 or computedafterwards. Such a graph may have nodes associated to panels and arcseach associated to the bend line separating the two adjacent panelsassociated to the nodes connected by the respective arc, as graph 500 ofFIG. 5. A cycle of the graph corresponds to a contemplated cycle of themethod. Moreover, the bend lines involved in said cycle are concurrent(i.e. they all intersect at a same point, e.g. considering theirinfinite support lines). The data defining the folded sheet objectfurther contains predetermined angles α_(i) between successive bendlines. In other words, given two “successive” bend lines (i.e.consecutive on the adjacency graph), thus associated to a same panel i(and separating said panel from a different other panel on each side ofthe panel), the angle α_(i) between the bend lines (e.g. on the side ofthe panel associated to the bend lines) is known. Said angle ispredetermined and fixed, thereby modeling rigidity of the panels.

Now, given such information on the folded sheet object, the methoddetermines at S20 a control law linking the angles β_(i) between theadjacent panels of the cycle, as a function of the predetermined anglesα_(i). The angles β_(i) are the angles between two adjacent panels (i.e.angles between two planes intersecting on the bend line separating theadjacent panels, e.g. taken on the side of the planes occupied by thepanels). At any time, the folded sheet object may be described, in termsof its position e.g. during folding, by the angles between all pairs ofadjacent panels. The method is discussed for the case of onecontemplated cycle. But as many control laws as there are cycles (whichmay generally be one or more) may be determined by the method, andcontrol laws linking other angles, i.e. not belonging to cycles, mayalso be determined by the method. The way several such control laws areorganized and the way to simulate the folding of the whole folded sheetobject is however out of the scope of the present discussion, as it maybe an implementation detail. The control law determined at S20 is anykind of data that tells how the angles β_(i) (of the contemplated cycle)relate all together. The control law thus corresponds to physicalfolding possibilities offered for a physical sheet corresponding to thefolding sheet object. The control law is provided as a function of theangles α_(i) such that the simulation of the folding necessarilycorresponds to what is physically possible.

For example, the control law may fix, given predetermined values for anumber of the angles β_(i) between pairs of the adjacent panels equal tothe degree of freedom of the cycle, values for the angles β_(i) betweenother pairs of the adjacent panels. This allows a fast and easysimulation.

Indeed, a cycle may have a degree of freedom, depending on the number ofpanels the cycle comprises and on the predetermined angles α_(i) betweensuccessive bend lines. A bend line is said to be folded when the anglebetween its two adjacent panels changes. The degree of freedom is thehighest number of bend lines (of the cycle) that may be foldedseparately, in the theoretical case that no constraint exists on thepanels except for the fact they are rigid and attached by the bend linesin a rotary way. For example, a cycle of four panels such as cyclescomprising panels 1, 2, 3 and 4 of sheet 400 of FIG. 4 or sheet 750 ofFIG. 7 has a degree of freedom equal to 1, as performing a folding forany pair of the adjacent panels physically leads the other panels of thecycle to move accordingly in order to accompany the folding (the fourpanels thereby forming one group). Now, if values for a number N of theangles β_(i) are provided, with N equal to such a degree of freedom, itphysically means that the other angles β_(i) have to take correspondingvalues (these values are fixed by the fact that the panels are rigid andare attached by the bend lines). The control law may solve that, by“fixing” such values (i.e. outputting the “fixed” values stemming fromthe physical configuration of a real sheet represented by the foldedsheet object). Referring back to FIG. 13, the simulation method may thensimply provide at S30 a number of values equal to the degree of freedom,such that the determining S40 may directly be performed with the controllaw. The case where fewer values of angles β_(i) than the degree offreedom are provided is detailed later. Simply put, the simulation mayset values for other angles β_(i) in any way so that the total number ofvalues reaches the degree of freedom, and the simulation may thendirectly use the control law of the example.

As exemplified later, the folding may comprise several (discrete)configurations. A folding configuration is a way of completing thefolding of the physical sheet, in the theoretical case that noconstraint exists on the panels except for the fact they are rigid andattached by the bend lines in a rotary way. Actually, as known from thepackaging industry, a same folded sheet object may be folded indifferent ways, depending, for a given degree of freedom, on thegeometry of the cycle, including the values of predetermined anglesα_(i) between successive bend lines. This variety of folding ways leadsto different results at the end of the physical folding (until panelsstick together and physically block the folding). These differentdiscrete results correspond to so-called folding configurations.Starting from a planar state, for example, given folded sheet object 400of FIG. 4, the folding may result in the configuration represented onFIG. 16, but it could also result in another configuration, representedon FIG. 22, as discussed later. This is because different strategies maybe followed to fold the cycle at point P.

As will be exemplified later, the control law may fix, givenpredetermined values for a number of the angles β_(i) between pairs ofthe adjacent panels equal to the degree of freedom of the cycle, a valuefor each angle β_(i) between other pairs of the adjacent panels, forany, several or each respective folding configuration. In other words,the control law may be adapted to the fact that there may be severalfolding configurations, and the control law may output angles for theangles β_(i) according to any requirements regarding the configurationsthat may be simulated. This allows any wanted simulation to be feasible.

Referring back to the simulation of the folding of FIG. 13, thesimulation may simulate any, several or all the folding configurations,that are allowed by the values provided at S30. Alternatively, the usermay select one or more configurations to simulate, for example byindicating virtual pressures to perform on at least one of the panels.Such virtual pressures may correspond to physical pressures applied by afolding machine on the physical sheet represented by the folded sheetobject, during a packaging process.

For example, the folded sheet object may be initially provided as aplanar sheet, e.g. having one or more cycles as defined above. Then theuser may activate a folding simulation function of the system. Then theuser may be asked to apply virtual pressures on panels of the cycles(s),thereby, for each cycle, defining initial values for a number N of theangles (β_(i)) between pairs of the adjacent panels (φ), with N inferioror equal to the degree of freedom of the cycle (N=1 in the case of acycle of four panels). For example a value corresponding to apredetermined angular delta (e.g. equal to 1°) is set for each angleβ_(i) between two adjacent panels for which a pressure is applied to oneof the panels (so-called “initial angle”). For example, one of the twopanels may be fixed, and the other one, to which a pressure is applied,rotated related to the fixed panel. Alternatively, such values may bedefined automatically, in any possible way. In any case, said initialvalues are provided at S30, and may already define one foldingconfiguration in this example. The simulation method may then, at thedetermining S40, determine the values for the other angles β_(i), asexplained earlier. Steps S30 and S40 may be repeated, each time addingthe predetermined angular delta to each initial angle. The addition ofthe predetermined angular delta is stopped once two adjacent panels arefolded such that one lies on the other. The folded sheet object may berepresented, e.g. displayed, throughout such iteration, so that thewhole folding procedure may be represented.

As can be seen, the goal of the presented methods is to solve complexcyclic folding with a control law, e.g. by using a closed form formulaas discussed later. Thanks to the control law, the folding of the foldedsheet object may be easily simulated as explained earlier, by giving avalue for some of the angles β_(i) at S30, and deducing at S40 thevalues of the other angles β_(i) thanks to the control law, possiblyautomatically.

As mentioned above, for the simulation, the user may be asked to provideauthentic design information to the system, at the beginning of theproviding S30, e.g.:

choose the initial folding orientation; and/or

if there are more than four bend lines, choose the controlled bendline(s) and define the angle(s) variation(s).

An example where folded sheet 400 of FIG. 4 is provided at S10 is nowdiscussed to illustrate the case of four concurrent bend lines and asmany adjacent panels.

FIG. 16 illustrates the three-dimensional shape 160 of folded sheetobject 400 of FIG. 4, used hereafter to illustrate the design method forthe cycle around point P (including panels 1, 2, 3 and 4). FIG. 17illustrates local shape 170 of panels 1, 2, 3 and 4 involved in thecycle in an intermediary position where panel 2 is not yet onto panel 1,like in FIG. 16.

The method of the example solves the problem of simulating a complexcyclic folding. The solution is to evaluate a mathematical functiongiven by the closed form formula. In other words, no numerical oriterative algorithm is needed, which makes the method simple, fast, andCPU-efficient. Furthermore, as discussed later the method mayadvantageously reuse the singularity analysis of the closed formformula. This is particularly efficient in packaging design because theplanar starting situation is in fact a singular situation from theequations point of view.

The closed form formula implemented by the control law is now discussedfor a 4-cycle such as the one of the example of FIG. 17.

The geometry of the 4-cycle may be entirely specified by setting anglesα_(i), i=1, . . . , 4 between consecutive bend lines that are providedat S10. FIG. 18 shows angles α_(i).

In this example, the cycle has four adjacent panels and the control lawcomprises the solution of an equation of the type F(ψ,φ)=0. In otherwords, the control law describes a process that outputs, givenpredetermined values for a one of the angles β_(i) between pairs of theadjacent panels, referred to as q), the degree of freedom of the cyclebeing equal to 1, values for the angles β_(i) between other pairs of theadjacent panels, including ψ, said process comprising solving (i.e.outputting the solution of) F(ψ,φ)=0 so as to first determine ψ, andthen other angles β_(i) in a straightforward way. Angles ψ and φ are theangles between successive pairs of the adjacent panels. In other words,given a predetermined order between panels (e.g. clockwise oranti-clockwise, given any normal), two successive pairs of adjacentpanels are likewise ordered, by the fact that they share a common panel(as they are successive). Referring to FIG. 17, with a normal vectorpointing in the direction of the reader and a clockwise order, pair ofadjacent panels {2,3} succeeds to pair of adjacent panels {1,2}. In theformula, φ is predetermined. Solving the formula thus provides ψ, andeventually the other angles β_(i) of the cycle. It is said that thesetwo other angles “stem” from said solution for this reason, as they area directly computed from ψ and φ, since the degree of freedom of thecycle is 1. It is noted that the chosen starting angles for the solutionare a matter of naming convention. The order between ψ and φ is also amatter of convention of the order around point P.

To sum up, given three consecutive panels i, j, k, let angle φ be theangle between panels i and j and let angle ψ be the angle between panelsj and k, as illustrated on FIG. 19. In a first step, the basic formulafor a 4-cycle is an implicit relationship F(ψ,φ)=0. This allows a simplesolution and thus a simple and fast simulation.

The equation (function F) may specifically be:

F(ψ, φ) = k₁ − k₂cos  ψ + k₃cos  ψ cos  φ + k₄cos  φ − sin  ψsinφwith:$k_{1} = \frac{{\cos \; \alpha_{1}\cos \; \alpha_{2}\cos \; \alpha_{4}} - {\cos \; \alpha_{3}}}{\sin \; \alpha_{2}\sin \; \alpha_{4}}$k₂ = cot  α₄sin  α₁ k₃ = cos  α₁ k₄ = cot  α₂sin  α₁

and α_(i(i=1 . . . 4)) are the predetermined angles between successivebend lines, where ψ is the angle between the pair of the adjacent panelscorresponding to α₁ and α₂, and φ is the angle between the pair of theadjacent panels corresponding to α₁ and α₃.

Solving F(ψ, φ)=0 with respect to angle ψ yields two closed formexpressions of angle ψ with respect to angles φ and α_(i), eachcorresponding to a different folding configuration:

$\psi^{+} = {\alpha + {\cos^{- 1}\left( \frac{c}{\sqrt{a^{2} + b^{2}}} \right)}}$$\psi^{-} = {\alpha - {\cos^{- 1}\left( \frac{c}{\sqrt{a^{2} + b^{2}}} \right)}}$

Where:

a=k ₂ −k ₃ cos φ

b=sin φ

c=−k ₁ −k ₄ cos φ

And where angle φ is defined by:

${\cos \; \alpha} = \frac{a}{\sqrt{a^{2} + b^{2}}}$${\sin \; \alpha} = \frac{b}{\sqrt{a^{2} + b^{2}}}$

In order to capture parameters dependency, ψ⁺ and φ⁻ are considered asfunctions ψ⁺(α₁, α₂, α₃, α₄, φ) and ψ⁻(α₁, α₂, α₃, α₄, φ). The formulasthat define these functions are valid as long as angles φ and α_(i) aresuch that c²≦a²+b². If angles φ and α_(i) are such that c²>a²+b², thereexists no angle ψ such that F(ψ,φ)=0. In such a case, the method mayoutput that the simulation is not feasible. If angles φ and α_(i) aresuch that c²=a²+b², then

$\frac{c}{\sqrt{a^{2} + b^{2}}} = {\pm 1}$

and ψ⁺=ψ⁻=α.

An example of the initial situation when solving the four bend linescomplex folding is now discussed.

In the context of packaging design, the starting situation is a flatcarton board sheet, meaning that for the folded sheet object:

α₁+α₂+α₃+α₄=2π

ψ₀=π

φ₀=π

Then, b=0 and c²=a², which makes ψ₀=π a double solution. This means thatwhen starting to fold, that is, when changing the value of angle φ₀=πinto a smaller value φ<π, the two motions φ

ψ⁺(α₁, α₂, α₃, α₄, φ) and φ

ψ⁻(α₁, α₂, α₃, α₄, φ) are possible, corresponding to a respectivefolding configuration. Consequently, the designer may choose onesolution by orienting the folding of one bend line, for example bydefining a virtual pressure on a panel adjacent to another panel defined(again by the user) to be fixed. It must be understood that, in thesimulation of one folding configuration, after the folding startedaccording to the user's choice, the solution is locally unique and themotion is well defined, as illustrated on FIG. 20.

FIGS. 21-22 illustrate this choice. FIG. 22 is the alternate foldingwhere panels 1 and 2 remain coplanar as well as panels 3 and 4. Thesingularity analysis of the planar initial configuration states that ifone α_(i) angle is larger than π, then the said configuration isisolated, meaning that the folding is impossible.

Panels positions may be computed by associating the following xyz axissystem to the 4-cycle. The origin of the axis system is the concurrentpoint of bend lines and the first vector of the axis system is the bendline u₀ shared by panels 1 and 2. Bend line v₀ is shared by panels 1 and4. Bend line w₀, is shared by panels 3 and 4. Bend line s₀ is shared bypanels 2 and 3. Axis system, bend lines vectors and panels areillustrated on FIG. 23.

At initial flat position, and considering that Σ_(i) α_(i)=2π, thecoordinates of vectors u₀, v₀, w₀, and s₀ are the following.

$u_{0} = \begin{pmatrix}1 \\0 \\0\end{pmatrix}$ $v_{0} = \begin{pmatrix}{\cos \; \alpha_{1}} \\{\sin \; \alpha_{1}} \\0\end{pmatrix}$ $w_{0} = \begin{pmatrix}{\cos \; \left( {\alpha_{1} + \alpha_{4}} \right)} \\{\sin \left( {\alpha_{1} + \alpha_{4}} \right)} \\0\end{pmatrix}$ $s_{0} = \begin{pmatrix}{\cos \left( {\alpha_{1} + \alpha_{4} + \alpha_{3}} \right)} \\{\sin \left( {\alpha_{1} + \alpha_{4} + \alpha_{3}} \right)} \\0\end{pmatrix}$

During the motion, panel number 1 is fixed, meaning that u₀ and v₀, donot depend on angle φ. Notation R(θ, u) is the rotation defined by angleθ and axis u. Then, as the angle φ changes from the initial value φ=π toa final value φ=φ₁, the corresponding motion of bend line w₀ is:

w(φ)=R(π−φ,v ₀)w ₀

meaning that bend line w₀ is rotated with angle it π−φ around axis v₀.Similarly, the corresponding motion of bend line s₀ is:

s(φ)=R(ψ^(ε)(φ)+π,u ₀)s ₀

meaning that bend line s₀ is rotated with angle ψ^(ε)(φ)+π around axisu₀. Symbol ε is + or − depending on the user's initial choice.

It should be noticed that the angle between vector w(φ) and vector s(φ)is equal to α₃ for any value of angle φ. This is because the rigidity ofeach panel is captured by functions ψ⁺ and ψ⁻.

FIG. 24 illustrates the useful rotations in a perspective view of theinitial flat situation. By decreasing angle φ from initial value φ₀=π toa target value φ₁ and by positioning bend lines according to theprevious process, positions of all four panels are completely definedfor each φε[φ₀, φ₁], as illustrated in FIGS. 25-29 which chronologicallyshow the folding for

${\alpha_{1} = \frac{3\pi}{4}},{\alpha_{2} = \frac{\pi}{4}},{\alpha_{3} = \frac{\pi}{2}},{\varphi_{0} = {{\pi \mspace{14mu} {and}\mspace{14mu} \varphi_{1}} = {\frac{\pi}{2}.}}}$

The previous process provides exact positioning of all panels. Inaddition, and for practical reasons, it might be useful to compute anglevalues between adjacent panels. Panels are numbered 1, 2, 3, 4 accordingto their respective bend line angles α₁, α₂, α₃, α₄.

By definition, angle β₁ is the angle between panels 4 and 1, β₂ is theangle between panels 1 and 2, β₃ is the angle between panels 2 and 3,and β₄ is the angle between panels 3 and 4, as illustrated in FIG. 30.According to previous section, β₁(φ)=φ is the input angle andβ₂(φ)=ψ^(ε)(φ). Angles β₃ and β₄ are computed as follows.

Firstly, normal vectors of panels 2, 3 and 4 are computed by using theirrespective bend lines. Bend lines of panel 2 are u₀ and s(φ), so normalvector n₂ of panel 2 is the normalized cross product

$n_{2} = \frac{{s(\varphi)} \times u_{0}}{{{s(\varphi)} \times u_{0}}}$

Bend lines of panel 3 are w(φ) and s(φ), so normal vector n₃ of panel 3is the normalized cross product:

$n_{3} = \frac{{w(\varphi)} \times {s(\varphi)}}{{{w(\varphi)} \times {s(\varphi)}}}$

Bend lines of panel 4 are v₀ and w(φ), so normal vector n₄ of panel 4 isthe normalized cross product:

$n_{4} = \frac{v_{0} \times {w(\varphi)}}{{v_{0} \times {w(\varphi)}}}$

Secondly, a panel is locally considered as an oriented half planelimited by a bend line. In order to capture which side of the half planeincludes the panel, a so-called “side vector” is needed. This sidevector is defined by the cross product of the normal vector of the panelwith the bend line direction oriented according to the local topology.The naming convention is that m_(ij) is the side vector of bend linemeasuring β_(i) with respect to panel j. Since each bend line is sharedby two panels, there are eight side vectors.

FIG. 31 illustrates all topological orientations on the top view of theinitial flat situation. Normal vectors point toward the reader.According to the definition, vectors m_(ij) are computed as follows.

m ₁₁ =n ₁×(−v ₀)

m ₁₄ =n ₄ ×v ₀

m ₂₁ =n ₁ ×u ₀

m ₂₂ =n ₂×(−u ₀)

m ₃₂ =n ₂ ×s(φ)

m ₃₃ =n ₃×(−s(φ))

m ₄₃ =n ₃ ×w(φ)

m ₄₄ =n ₄×(−w(φ))

Now, cosines of angles β₃ and β₄ are the scalar products of adjacentside vectors, that is:

cos(β₃)=(m ₃₂ ,m ₃₃)

cos(β₄)=(m ₄₃ ,m ₄₄)

In order to determine the signs of sinus values, the so-called “sector”is introduced. By definition the sector of two adjacent panels is theregion of 3D space that includes the normal vectors of both panels.Angles β_(i) precisely define these sectors. In fact the so-called“sectors sharpness's” σ_(i), i=1, . . . , 4 are useful. The sectorsharpness is the sign of the scalar product of the sum of side vectorswith the sum of normal vectors. The sharpness's of interest are:

σ₃=Sign(m ₃₂ +m ₃₃ ,n ₂ +n ₃)

σ₄=Sign(m ₄₃ +m ₄₄ ,n ₃ +n ₄)

By construction, if this scalar product is positive (respectivelynegative), the sector is sharp (resp. non sharp). FIG. 32-33respectively illustrate a sharp sector for angle β₃ and a non-sharpsector for angle β₄. The scalar product vanishes either when the sumΣn_(i) of normal vectors vanishes or when the sum Σm_(ij) of sidevectors vanishes. When the sum of side vectors vanishes, the sector isflat and the angle β is equal to π. When the sum of normal vectorsvanishes, the sector is either full and angle β is equal to 2π, or thesector is knife-shaped and angle β is equal to 0, which is equivalent.FIGS. 34-36 illustrate degenerate sectors. Finally, if the sector ofangle β_(i) is sharp, then sin(β_(i))>0. If the sector of angle β_(i) isnon-sharp, then sin(β_(i))<0. Tables I and II gather all the results forcomputing respectively angles β₃ and β₄, which ends the determinationS40 of angles between adjacent panels.

TABLE I computation of angle β₃ β₃ σ₃ > 0 cos⁻¹ <m₃₂, m₃₃> σ₃ < 0 2π −cos⁻¹ <m₃₂, m₃₃> σ₃ = 0 n₂ + n₃ = 0 0 2π m₃₂ + m₃₃ = 0 π

TABLE II computation of angle β₄ β₄ σ₄ > 0 cos⁻¹ <m₄₃, m₄₄> σ₄ < 0 2π −cos⁻¹ <m₄₃, m₄₄> σ₄ = 0 n₃ + n₄ = 0 0 2π m₄₃ + m₄₄ = 0 π

Obviously, according to the geometry provided at S10, the control lawmay thus indicate if an objective of angles β_(i) is possible or not,and determined the remaining angles β_(i) only when it is possible.

The general case when the cycle has more than four panels (four panelsor more) is now discussed.

In such a case, the design method may determine at S20 the control lawbased on the theoretical developments for determining the control law inthe case the cycle has four panels. Namely, the determining S20 maycomprise partitioning the panels into four groups. In other words, thepanels are gathered into groups of at least one panel (depending on thetotal number of panels of the cycle). Then, for each group whichcomprises at least two panels, the method may determine a virtual panelthat forms a cycle with the panels of the group (i.e. the virtual panelis a theoretical panel that would form a cycle of adjacent panels withthe panels of the group), thereby forming a virtual cycle of fouradjacent panels (virtual or not virtual) that each correspond to arespective group. In other words, the cycle of more than four panels isassimilated to a cycle of four panels. As a consequence, the determiningS20 may easily end by defining equation F defined for the virtual cycle,just like what is performed in the case of four panels (explainedabove), and the control law determined at S20 may then simply comprisethe solution of said equation F.

The relevant point in such a generalization of the four panel case isthat the virtual panels are not rigid, unlike the real initial panels ofthe folded sheet object. Thus, the angles between successive bend linesof the virtual cycle are not necessarily fixed. However, for the purposeof solving F for the virtual cycle, said angles are fixed (thus, theangles between the panels forming a cycle with a virtual panel are setto a determined value at a given time of the simulation of the folding).How to set such angles between the panels is explained later. It isnoted that in the implementation of the algorithm, recursion may be usedto bring the problem of more than four panels to the problem of fourpanels, according to the creation of virtual panels discussed here.

The generalization is now discussed with the specific case of afive-panel cycle.

When dealing with a five-panel cycle, the principle is to define anequivalent four-panel cycle and to reuse the previous formulas. This isdone by replacing two adjacent panels by an equivalent panel asillustrated in FIGS. 37-39. Which pair of adjacent panels should bereplaced by an equivalent panel depends on the overall folding strategy,which may involve the user's choice. It must be understood that theso-called “equivalent panel” or “virtual panel” is an artifact forreusing the closed form expressions of ψ⁺(·) and ψ⁻(·). It may not haveany physical meaning.

FIG. 37 shows the initial five-cycle, where α_(i), i=1, . . . , 5 areangle between consecutive bend lines. FIG. 38 drawing displays theequivalent panel together with the pair (2,3) of adjacent panels to bereplaced. Angle 6 is the angle between bend lines of the equivalentpanel. FIG. 39 shows the equivalent four-cycle.

In order to reuse the four-cycle formula, it is necessary to compute theangle of the equivalent panel according to those of the replaced panels.FIGS. 40-41 illustrate adjacent panels ABC and ADC sharing bend line ACand the equivalent panel ABD. Let γ=

and β=

be the respective angles between the bend lines of the two adjacentpanels and let θ be the angle between their respective planes, asillustrated in FIGS. 42-43.

Then, the angle δ=

of the equivalent panel is given by the following formula:

δ(β,γ,θ)=cos⁻¹(cos β cos γ+sin β sin γ cos θ)

Now, when the cycle has more than four panels and a degree of freedomhigher or equal to 2 (which is the case of the example now discussed),at least one angle between a pair of adjacent panels may be controlledby a variation law depending on the angle between another pair ofadjacent panels. A variation law is thus a function that links twoangles β_(i) together, by putting the value of one under the control ofthe value of the other. Such a variation law may simplify the task ofthe user when simulating the folding in a case of more than four panels.The variation law may be deduced from physical or machining constraintsof the folding.

In the example, during the folding motion, angle θ is controlled by avariation law depending on the driving angle φ. This variation lawdepends on the folding strategy and may involve the user's decision.This means that angle δ between the two bending lines of the replacingpanel is allowed to change during the motion. Collecting all data in thesame drawing yields to FIG. 44.

By choosing φ as the driving angle and by noting the variation lawθ=θ(φ), the angle δ=δ(α₂, α₃, θ(φ)) is related to according to theprevious formula. Finally, angle ψ is given by the following formula,where ε∈{+, −} depends on the initial choice,

ψ=ψ^(ε)(α₁,δ(α₂,α₃,θ(φ)),α₄,α₅,φ)

Generalization to an arbitrary number of bend lines is possible bysuccessively combining pairs of adjacent panels until the resultingnumber of panels is four.

For example, given a 6-cycle α₁, . . . , α₆, suppose that angle φcontrols the angle between adjacent panels 1 and 6. Let (1,2,3,4,5,6) bethe notation for the sequence of adjacent panels. According to theuser's defined folding strategy, panels 2 and 3 are replaced by anequivalent panel, noted (2,3). This requires a variation law θ₁=θ₁(φ)for controlling the angle between panels 2 and 3 and this introducesangle δ₁(α₂, α₃, θ₁(φ)) between the bend lines of the equivalent panel(2,3). This reduces the number of adjacent panels from six to five, andthe new sequence is noted (1, (2,3), 4,5,6). Then, following the userdefined folding strategy, adjacent panels 4 and 5 are replaced by anequivalent panel, noted (4,5), thus requiring another variation lawθ₂=θ₂(φ) for controlling the angle between panels 4 and 5 andintroducing angle δ₂ (α₄, α₅, θ₂(φ)) between bend lines of theequivalent panel (4,5). This is represented on FIGS. 45-46.

Now the number of adjacent panels is four, the new sequence is noted (1,(2,3), (4,5), 6), allowing the following formula for controlling theangle between panel 1 and panel (2,3).

ψ=ψ^(ε)(α₂,α₃,θ₁(φ)),δ₂(α₄,α₅,θ₂(φ)),α₆,φ)

The process of replacing a pair of adjacent panels by an equivalentpanel may involve another equivalent panel. For example, after panels 2and 3 are replaced by an equivalent panel (2,3) like in the previousexample, meaning that the sequence is (1, (2,3), 4,5,6), the foldingstrategy could replace adjacent panels (2,3) and 4 by an equivalentpanel, noted ((2,3), 4), as shown on FIGS. 47-48.

The corresponding sequence is (1, (2,3), 4,5,6) and the formula forangle ψ is

ψ=ψ^(ε)(α₁,δ₂(δ₁(α₂,α₃,θ₁(φ)),α₄,θ₂(φ)),α₅,α₆,φ)

Angle ψ drives the angle between panel 1 and panel ((2,3), 4).

The teachings of all patents, published applications and referencescited herein are incorporated by reference in their entirety.

While this invention has been particularly shown and described withreferences to example embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims.

What is claimed is:
 1. Computer-implemented method for designing afolded sheet object, comprising the steps of: providing (S10) panelsseparated by bend lines, including at least four adjacent panels forminga cycle and separated by concurrent bend lines, with predeterminedangles (α_(i)) between successive bend lines; and determining (S20) acontrol law linking the angles (β_(i)) between the adjacent panels ofthe cycle, as a function of the predetermined angles (α_(i)) betweensuccessive bend lines.
 2. The method of claim 1, wherein the control lawfixes, given predetermined values for a number of the angles (β_(i))between pairs of the adjacent panels (φ) equal to the degree of freedom(1) of the cycle, values for the angles (β_(i)) between other pairs ofthe adjacent panels (ψ).
 3. The method of claim 2, wherein, when thecycle has four adjacent panels, the control law comprises the solutionof an equation of the type F(ψ,φ)=0, where ψ and φ are the anglesbetween successive pairs of the adjacent panels and φ is predetermined,the angles (β_(i)) between the two other pairs of adjacent panels of thecycle stemming from said solution.
 4. The method of claim 3, wherein theequation is:F(ψ, φ) = k₁ − k₂cos  ψ + k₃cos  ψcos φ + k₄cos  φ − sin  ψ sin  φwith:$k_{1} = \frac{{\cos \; \alpha_{1}\cos \; \alpha_{2}\cos \; \alpha_{4}} - {\cos \; \alpha_{3}}}{\sin \; \alpha_{2}\sin \; \alpha_{4}}$k₂ = cot  α₄sin  α₁ k₃ = cos  α₁ k₄ = cot  α₂sin  α₁ andα_(i(i=1 . . . 4)) are the predetermined angles (α_(i)) betweensuccessive bend lines, where ψ is the angle between the pair of theadjacent panels corresponding to α₁ and α₂, and φ is the angle betweenthe pair of the adjacent panels corresponding to α₁ and α₃.
 5. Themethod of claim 3, wherein, when the cycle has more than four panels,determining the control law comprises: partitioning the panels into fourgroups; and for each group which comprises at least two panels,determining a virtual panel that forms a cycle with the panels of thegroup, thereby forming a virtual cycle of four adjacent panels that eachcorrespond to a respective group; and the control law comprises thesolution of equation F defined for the virtual cycle.
 6. The method ofclaim 5, wherein, when the cycle has more than four panels and a degreeof freedom higher or equal to 2, at least one angle (θ) between a pairof adjacent panels is controlled by a variation law depending on theangle (φ) between another pair of adjacent panels.
 7. A folded sheetobject obtainable by the method of claim
 1. 8. A data file storing thefolded sheet object of claim
 7. 9. A computer-implemented method forsimulating the folding of the folded sheet object of claim 7, comprisingthe steps of: providing (S30) predetermined values for a number of theangles (β_(i)) between pairs of the adjacent panels (φ); determining(S40) values of the other angles (β_(i)) between pairs of the adjacentpanels (ψ) with the control law.
 10. A computer program productcomprising: a data storage medium having recorded thereon program codecausing a computer to design a folded sheet object; and the program codeincluding instructions causing the computer to: provide (S10) panelsseparated by bend lines, including at least four adjacent panels forminga cycle and separated by concurrent bend lines, with predeterminedangles (α_(i)) between successive bend lines; and determine (S20) acontrol law linking the angles (β_(i)) between the adjacent panels ofthe cycle, as a function of the predetermined angles (α_(i)) betweensuccessive bend lines.
 11. The computer program product of claim 10wherein the program code further includes instructions causing thecomputer to simulate folding of the folded sheet object by: providing(S30) predetermined values for a number of the angles (β_(i)) betweenpairs of the adjacent panels (φ); determining (S40) values of the otherangles (β_(i)) between pairs of the adjacent panels (ψ) with the controllaw.
 12. A CAD system comprising: a memory; and a processor coupled tothe memory and a graphical user interface, the memory having recordedthereon program code which when executed by the processor: provides(S10) panels separated by bend lines, including at least four adjacentpanels forming a cycle and separated by concurrent bend lines, withpredetermined angles (α_(i)) between successive bend lines; anddetermines (S20) a control law linking the angles (β_(i)) between theadjacent panels of the cycle, as a function of the predetermined angles(α_(i)) between successive bend lines.
 13. The CAD system as claimed inclaim 12 wherein the control law fixes, given predetermined values for anumber of the angles (β_(i)) between pairs of the adjacent panels (φ)equal to the degree of freedom (1) of the cycle, values for the angles(β_(i)) between other pairs of the adjacent panels (ψ).
 14. The CADsystem as claimed in claim 13 wherein, when the cycle has four adjacentpanels, the control law comprises the solution of an equation of thetype F(ψ,φ)=0, where ψ and φ are the angles between successive pairs ofthe adjacent panels and φ is predetermined, the angles (β_(i)) betweenthe two other pairs of adjacent panels of the cycle stemming from saidsolution.
 15. The CAD system as claimed in claim 14 wherein the equationis:F(ψ, φ) = k₁ − k₂cos  ψ + k₃cos  ψcos φ + k₄cos  φ − sin  ψ sin  φwith:$k_{1} = \frac{{\cos \; \alpha_{1}\cos \; \alpha_{2}\cos \; \alpha_{4}} - {\cos \; \alpha_{3}}}{\sin \; \alpha_{2}\sin \; \alpha_{4}}$k₂ = cot  α₄sin  α₁ k₃ = cos  α₁ k₄ = cot  α₂sin  α₁ andα_(i(i=1 . . . 4)) are the predetermined angles (α_(i)) betweensuccessive bend lines, where ψ is the angle between the pair of theadjacent panels corresponding to α₁ and α₂, and φ is the angle betweenthe pair of the adjacent panels corresponding to α₁ and α₃.
 16. The CADsystem as claimed in claim 14 wherein, when the cycle has more than fourpanels, determining the control law comprises: partitioning the panelsinto four groups; and for each group which comprises at least twopanels, determining a virtual panel that forms a cycle with the panelsof the group, thereby forming a virtual cycle of four adjacent panelsthat each correspond to a respective group; and the control lawcomprises the solution of equation F defined for the virtual cycle. 17.The CAD system as claimed in claim 16 wherein, when the cycle has morethan four panels and a degree of freedom higher or equal to 2, at leastone angle (θ) between a pair of adjacent panels is controlled by avariation law depending on the angle (φ) between another pair ofadjacent panels.